Signal and Systems Chapter 8: Modulation Complex Exponential - - PowerPoint PPT Presentation

Signal and Systems Chapter 8: Modulation

• Complex Exponential Amplitude Modulation
• Sinusoidal AM
• Demodulation of Sinusoidal AM
• Single-Sideband (SSB) AM
• Frequency-Division Multiplexing
• Superheterodyne Receivers
• AM with an Arbitrary Periodic Carrier
• Pulse Train Carrier and Time-Division Multiplexing
• Sinusoidal Frequency Modulation
• DT Sinusoidal AM
• DT Sampling, Decimation, and Interpolation

The Concept of Modulation

 Why?  More efficient to transmit signals at higher frequencies  Transmitting multiple signals through the same medium using different

carriers

 Transmitting through “channels” with limited pass-bands  Others…  How?  Many methods  Focus here for the most part on Amplitude Modulation (AM)

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 2

Amplitude Modulation (AM) of a Complex Exponential Carrier

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 3

carrier frequency

( ) , ( ) ( ) 1 ( ) ( ) ( ) 2 1 ( ) 2 ( ) 2 ( ( ))

c c

j t c j t c c

c t e y t x t e Y j X j C j X j X j

 

                     

Demodulation of Complex Exponential AM

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 4

cos sin

c

j t c c

e t j t



    

Corresponds to two separate modulation channels (quadratures) with carriers 90˚ out of phase.

Sinusoidal AM

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 5

1 ( ) ( ) { ( ) ( )} 2 1 1 ( ( )) ( ( )) 2 2

c c c c

Y j X j X j X j                        

Drawn assuming

c M

  

Synchronous Demodulation of Sinusoidal AM

 Suppose θ= 0 for now, ⇒

Local oscillator is in phase with the carrier.

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 6

Synchronous Demodulation in the Time Domain

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2 High-Frequency Signals filterd out by the LRF

1 ( ) ( )cos ( )cos ( )cos2 2 Then ( ) ( ) Now suppose there is phase difference, i.e. 0, then ( ) ( )cos( ) ( )cos cos( )

c c c c c c

w t y t t x t t x t t r t x t w t y t t x t t t                  

HF signal

1 1 ( )cos ( )(cos(2 )) 2 2 Now ( ) ( )cos Two special cases: 1) 2, the local oscillator is 90 out of phase with the carrier, ( ) 0,signal unrecoverable. 2) ( ) slowly var

c

x t x t t r t x t r t t                  ying with time, ( ) cos[ ( )] ( ), time-varying "gain". r t t x t    

Synchronous Demodulation (with phase error) in the Frequency Domain

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 8

Demodulating signal has phase difference w.r.t. the modulating si cos( gnal )

1 1 2 2 ( ) ( )

c c

j t j t j j c j j c c

t e e e e F e e

     

 

        

  

 

      Again, the low-frequency signal ( )when 2.

M

     

Alternative: Asynchronous Demodulation

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 9

Assume ,so signal envelope looks like ( )

c M

x t   

DSB/SC (Double Side Band, Suppressed Carrier) DSB/WC (Double Side Band, With Carrier) A A    

Asynchronous Demodulation (continued)Envelope Detector

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 10

In order for it to function properly, the envelope function must be positive for all time, i.e. A+ x(t) > 0 for all t. Demo: Envelope detection for asynchronous demodulation. Advantages of asynchronous demodulation: — Simpler in design and implementation. Disadvantages of asynchronous demodulation: — Requires extra transmitting power [Acosωct]2to make sure A+ x(t) > 0 ⇒Maximum power efficiency = 1/3 (P8.27)

Double-Sideband (DSB) and Single- Sideband (SSB) AM

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 11

Since x(t) and y(t) are real, from Conjugate symmetry both LSB and USB signals carry exactly the same information. DSB, occupies 2ωMbandwidth in ω> 0 Each sideband approach only occupies ωM bandwidth in ω> 0

Single Sideband Modulation

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 12

Can also get SSB/SC or SSB/WC

Frequency-Division Multiplexing (FDM)

 (Examples: Radio-station signals and analog cell phones)

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 13

All the channels can share the same medium.

FDM in the Frequency-Domain

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Demultiplexing and Demodulation

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ωa needs to be tunable

 Channels must not overlap ⇒Bandwidth Allocation  It is difficult (and expensive) to design a highly selective

band-pass filter with a tunable center frequency

 Solution –Superheterodyne Receivers

The Superheterodyne Receiver

 Operation principle:  Down convert from ωc to ωIF, and use a coarse tunable BPF for the front

• end. (FCC: Federal Communications Commission)

 Use a sharp-cutoff fixed BPF at ωIF to get rid of other signals.

Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 16

AM with an Arbitrary Periodic Carrier

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C(t) – periodic with period T, carrier frequency ωc = 2π/T Remember: periodic in t discrete in ω

( ) 2 ( )

k c k

C j a k     

 

 



𝑏𝑙 = 1 𝑈 𝑔𝑝𝑠 𝑗𝑛𝑞𝑣𝑚𝑡𝑓 𝑢𝑠𝑏𝑗𝑜

1 ( ) ( )* ( ) ( )* ( ) 2

k c k

Y j X j C j X j a k        

 

  



( ( ))

k c k

a X j k  

 

 



⇓

Modulating a (Periodic) Rectangular Pulse Train

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 18

) ( ). ( ) ( t c t x t y 

Modulating a Rectangular Pulse Train Carrier, cont’d

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 19

( ) 2 ( )

k c k

C j a k     

 

 



𝑏𝑜𝑒 𝑏0 = 𝛦 𝑈 , 𝑏𝑙 = sin 𝑙𝜕𝑑 𝛦 2 𝜌𝑙 For rectangular pulse

1 ( ) ( )* ( ) 2 Y j X j C j     

Drawn assuming: 𝜕𝑑 > 2𝜕𝑁 Nyquist rate is met

Observations

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 20

1) We get a similar picture with any c(t) that is periodic with period T 2) As long as ωc= 2π/T > 2ωM, there is no overlap in the shifted and scaled replicas of X(jω). Consequently, assuming a0≠0: x(t) can be recovered by passing y(t) through a LPF 3) Pulse Train Modulation is the basis for Time-Division Multiplexing Assign time slots instead of frequency slots to different channels, e.g. AT&T wireless phones 4) Really only need samples{x(nT)} when ωc> 2ωM⇒Pulse Amplitude Modulation

Sinusoidal Frequency Modulation (FM)

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)) ( cos( ) ( t A t y  

Amplitude fixed

Phase modulation: 𝜄 𝑢 = 𝜕𝑑𝑢 + 𝜄0 + 𝑙𝑞𝑦 𝑢 Frequency modulation:

𝑒𝜄 𝑒𝑢 = 𝜕𝑑 + 𝑙𝑔𝑦(𝑢)

Instantaneous ω

X(t) is signal To be transmitted

Sinusoidal FM (continued)

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 22

 Transmitted power does not depend on x(t): average power = A2/2  Bandwidth of y(t) can depend on amplitude of x(t)  Demodulation

a) Direct tracking of the phase θ(t) (by using phase-locked loop) b) Use of an LTI system that acts like a differentiator H(jω) —Tunable band-limited differentiator, over the bandwidth of y(t)

/

( ) ( ) ( ( )) sin ( )

c f d dt

dy t u t w k x t A t dt



    

𝐼 𝑘𝜕 ≅ 𝑘𝜕 ⇓ …looks like AM envelope detection

DT Sinusoidal AM

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 23

Multiplication ↔Periodic convolution Example#1:

[ ]

c

j n

c n e   ( ) 2 ( 2 )

j c k

c e k



    

 

   



1 ( ) ( ) ( ) 2

j j j

Y e X e C e

  

  

Example#2: Sinusoidal AM

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 24

[ ] cos

c

c n n   ( ) ( 2 ) ( 2 )

j c c k

C e k k



        

 

           



0 and 2

c M c M c M

             / 2

M c M M

π          

1 ( ) ( ) ( ) 2

j j j

Y e X e C e

  

  

i.e., No overlap of shifted spectra Drawn assuming:

Example #2 (continued):Demodulation

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 25

Possible as long as there is no

• verlap of shifted replicas of

X(ejω):

i.e., and 2

c M c M c M M c M

                    

1 ( ) ( ) ( ) 2

j j j

W e Y e C e

  

  

Example #3: An arbitrary periodic DT carrier

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 26

2 /

2 [ ] [ ],

jk N k c k N

c n a e c n N N



 

 

   



2 ( ) 2

j k k

k C e a N



   

 

       



1 ( ) ( ) ( ) 2

j j j

Y e X e C e

  

  

1

1 2 ( )* 2 2

N j k k

k X e a N



    

 

       



1 ( 2 / )

( )

N j k N k k

a X e

    

 

• Periodic convolution
• Regular convolution

Example #3 (continued):

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 27

DT Sampling

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 28

Motivation: Reducing the number of data points to be stored or transmitted, e.g. in CD music recording.

DT Sampling (continued)

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 29

2 2 [ ] [ ] ( ) ( ),

j s s k k

p n n kN P e k N N



      

   

     

 



  

  

k p

kN n kN x n p n x n w ] [ ] [ ] [ ]. [ ] [ 

Note: 𝑦𝑞 𝑜 = 𝑦 𝑜 , 𝑗𝑔 𝑜 𝑗𝑡 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 𝑛𝑣𝑚𝑢𝑗𝑞𝑚𝑓 𝑝𝑔 𝑂 0, 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓

⇒ Pick one out of N

1 ( )

1 1 ( ) ( ) ( ) ( ) 2

s

N j k j j j p

X e X e P e X e N

    



 

  



• periodic with period 𝜕𝑡 =

2𝜌 𝑂

DT Sampling Theorem

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 30

We can reconstruct x[n] if ωs= 2π/N > 2ωM Drawn assuming ωs > 2ωM Nyquist rate is met ⇒ ωM< π/N Drawn assuming ωs < 2ωM Aliasing!

Decimation — Downsampling

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 31

xp[n] has (n-1) zero values between nonzero values: Why keep them around? Useful to think of this as sampling followed by discarding the zero values

Illustration of Decimation in the Time-Domain (for N= 3)

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 32

Decimation in the Frequency Domain

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 33

( ) [ ] ( [ ] [ ])

j jk b b b p k

X e x k e x k x kN

    

 



[ ] Let n kN or k n/N

jk p k

x kN e

   

  



( / ) an integer multiple of N

[ ]

j n N p n

x n e

   





( / ) p

[ ] (Since x [n kN] 0)

j N n p n

x n e

   

  



= 𝑌𝑞 𝑓𝑘

𝜕 𝑂

Squeeze in time Expand in frequency

• Still periodic with period 2π

since Xp(ejω) is periodic with 2π/N

Illustration of Decimation in the Frequency Domain

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 34

The Reverse Operation: Upsampling(e.g.CD playback)

Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 35